Stability of Force-Free Magnetic Fields

Abstract

A formalism for treating the stability of force-free magnetic fields ($\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathrm{B}=\ensuremath{\alpha}\mathrm{B}$) based on the energy principle of Bernstein et al. is derived. An example is given to shown that force-free fields with constant $\ensuremath{\alpha}$ may be unstable, though previous papers give arguments in favor of the stability of such fields. It is found that the cylindrically symmetric force-free field given by Lundquist is unstable if and only if $\ensuremath{\alpha}{r}_{0}$ (where ${r}_{0}$ is the radius of the cylinder) exceeds the critical value 3.176. The unstable displacements have small growth rates. They are of the screw or kink type, their wavelengths along the axis having a minimum of about seven times the critical cylinder radius.The results are applied to the arms of spiral galaxies, correcting Trehan's statement that some force-free fields with constant $\ensuremath{\alpha}$ have a stabilizing effect on gravitational instability.